How seamless does this need to be? Cuts along the wood's grain can often be glued back together almost undetectably, if you're careful.
Bent lamination could be done, but the connection to face and back would not be seamless -- grain wouldn't match. There would also be a grain mismatch seam where the laminate ends meet, unless you are careful and/or lucky, though that could be on the bottom of the box.
Hollowing out a cavity in a solid block doesn't need CNC, just careful work with drills (Forstner and similar bits are good for hogging out most of the waste) and/or chisels. A drill press is useful for controlling drilling depth.
Rounding corners is a generally matter of sawing away most of the waste, then refining the curve with rasps, files, sandpaper, scrapers, and the like, unless you go back to considering bent lamination.
Note that if you plan on painting this, all the issues with matching grain go away and it becomes a matter of how well you smoothe the glue joints.
You might want to soften the sharp edges at front and back. Or might not.